System of Equations Applications

Question

A 1400-seat theater sells two types of tickets for a concert. Premium seats sell for $30 each and regular seats sell for $20 each. At one event $34,060 was collected in ticket sales with 10 seats left unsold. How many of each type of ticket was sold?

Answer 1

Hello, I would love to explain you solution for this problem!

Let X be amount for tickets sold for 30$ and Y is amount for tickets for 20$.

We know that the theater has 1400 seats but there were 10 seats unsold. That means that 1400-10=1390 seats were sold. So we have our first equation of total amount of seats:

X+Y = 1390

Now we need the second equation. If X tickets were sold for 30$, the total cost that theater received for X tickets is 30X. If Y tickets were sold for 20$, the total cost that theater received for Y tickets is 20Y. We also know that "$34,060 was collected in ticket sales ". Here our second equation:

30X + 20Y = 34060

Now we have system of equations:

X+Y = 1390

30X + 20Y = 34060

Lets express Y in terms of X:

Y=1390-X

30X + 20Y = 34060

Now substitute Y into the second equation:

Y=1390-X

30X + 20(1390-X) = 34060

Solve second equation for X. First open brackets:

30X + 20*1390-20X = 34060

30X + 27800 - 20X = 34060

Leave all Xs on the left side and bring everything else to the right side.

30X - 20X = 34060 - 27800

Collect terms on both sides

10X = 6260

Divide both sides by 10

X=626

Now we know X. We can substitute is into first equation to get Y.

Y=1390 - X= 1390 - 626= 764

ANSWER: There were 626 premium seats and 764 regular seats sold.

I hope this helps! Be free to ask questions!